TRIGONOMETRY http math la asu edutdalesanmat 170TRIGONOMETRY ppt
- Slides: 23
TRIGONOMETRY http: //math. la. asu. edu/~tdalesan/mat 170/TRIGONOMETRY. ppt
Angles, Arc length, Conversions Angle measured in standard position. Initial side is the positive x – axis which is fixed. Terminal side is the ray in quadrant II, which is free to rotate about the origin. Counterclockwise rotation is positive, clockwise rotation is negative. Coterminal Angles: Angles that have the same terminal side. 60°, 420°, and – 300° are all coterminal. Degrees to radians: Multiply angle by Radians to degrees: Multiply angle by Note: 1 revolution = 360° = 2π radians. Arc length = central angle x radius, or Note: The central angle must be in radian measure. radians
Right Triangle Trig Definitions B c a C • • • b A sin(A) = sine of A = opposite / hypotenuse = a/c cos(A) = cosine of A = adjacent / hypotenuse = b/c tan(A) = tangent of A = opposite / adjacent = a/b csc(A) = cosecant of A = hypotenuse / opposite = c/a sec(A) = secant of A = hypotenuse / adjacent = c/b cot(A) = cotangent of A = adjacent / opposite = b/a
Special Right Triangles 30° 45° 2 1 1 60° 1 45°
Basic Trigonometric Identities Quotient identities: Even/Odd identities: Even functions Odd functions Reciprocal Identities: Pythagorean Identities: Odd functions
All Students Take Calculus. Quad III cos(A)<0 sin(A)>0 tan(A)<0 sec(A)<0 csc(A)>0 cot(A)<0 cos(A)>0 sin(A)>0 tan(A)>0 sec(A)>0 csc(A)>0 cot(A)>0 cos(A)<0 sin(A)<0 tan(A)>0 sec(A)<0 csc(A)<0 cot(A)>0 cos(A)>0 sin(A)<0 tan(A)<0 sec(A)>0 csc(A)<0 cot(A)<0 Quad IV
Unit circle • • Radius of the circle is 1. x = cos(θ) y = sin(θ) Pythagorean Theorem: This gives the identity: Zeros of sin(θ) are where n is an integer. Zeros of cos(θ) are where n is an integer.
Graphs of sine & cosine • • • Fundamental period of sine and cosine is 2π. Domain of sine and cosine is Range of sine and cosine is [–|A|+D, |A|+D]. The amplitude of a sine and cosine graph is |A|. The vertical shift or average value of sine and cosine graph is D. • The period of sine and cosine graph is • The phase shift or horizontal shift is
Sine graphs y = sin(x) y = 3 sin(x) y = sin(x) + 3 y = sin(3 x) y = sin(x – 3) y = sin(x/3) y = 3 sin(3 x-9)+3 y = sin(x)
Graphs of cosine y = cos(x) + 3 y = 3 cos(x) y = cos(3 x) y = cos(x – 3) y = cos(x/3) y = 3 cos(3 x – 9) + 3 y = cos(x)
Tangent and cotangent graphs • Fundamental period of tangent and cotangent is π. • Domain of tangent is where n is an integer. • Domain of cotangent where n is an integer. • Range of tangent and cotangent is • The period of tangent or cotangent graph is
Graphs of tangent and cotangent y = tan(x) Vertical asymptotes at y = cot(x) Verrical asymptotes at
Graphs of secant and cosecant y = sec(x) Vertical asymptotes at Range: (–∞, – 1] U [1, ∞) y = cos(x) y = csc(x) Vertical asymptotes at Range: (–∞, – 1] U [1, ∞) y = sin(x)
Inverse Trigonometric Functions and Trig Equations Domain: [– 1, 1] Range: 0 < y < 1, solutions in QI and QII. – 1 < y < 0, solutions in QIII and QIV. Domain: [– 1, 1] Range: [0, π] 0 < y < 1, solutions in QI and QIV. – 1< y < 0, solutions in QII and QIII. Domain: Range: 0 < y < 1, solutions in QI and QIII. – 1 < y < 0, solutions in QII and QIV.
Trigonometric Identities Summation & Difference Formulas
Trigonometric Identities Double Angle Formulas
Trigonometric Identities Half Angle Formulas The quadrant of determines the sign.
Law of Sines & Law of Cosines Law of sines Use when you have a complete ratio: SSA. Law of cosines Use when you have SAS, SSS.
Vectors • A vector is an object that has a magnitude and a direction. • Given two points P 1: and P 2: on the plane, a vector v that connects the points from P 1 to P 2 is v= i+ j. • Unit vectors are vectors of length 1. • i is the unit vector in the x direction. • j is the unit vector in the y direction. • A unit vector in the direction of v is v/||v|| • A vector v can be represented in component form by v = vxi + vyj. • The magnitude of v is ||v|| = • Using the angle that the vector makes with x-axis in standard position and the vector’s magnitude, component form can be written as v = ||v||cos(θ)i + ||v||sin(θ)j
Vector Operations Scalar multiplication: A vector can be multiplied by any scalar (or number). Example: Let v = 5 i + 4 j, k = 7. Then kv = 7(5 i + 4 j) = 35 i + 28 j. Dot Product: Multiplication of two vectors. Let v = vxi + vyj, w = wxi + wyj. Example: Let v = 5 i + 4 j, w = – 2 i + 3 j. v · w = (5)(– 2) + (4)(3) = – 10 + 12 = 2. v · w = vxwx + vywy Alternate Dot Product formula v · w = ||v||||w||cos(θ). The angle θ is the angle between the two vectors. v θ w Two vectors v and w are orthogonal (perpendicular) iff v · w = 0. Addition/subtraction of vectors: Add/subtract same components. Example Let v = 5 i + 4 j, w = – 2 i + 3 j. v + w = (5 i + 4 j) + (– 2 i + 3 j) = (5 – 2)i + (4 + 3)j = 3 i + 7 j. 3 v – 2 w = 3(5 i + 4 j) – 2(– 2 i + 3 j) = (15 i + 12 j) + (4 i – 6 j) = 19 i + 6 j. ||3 v – 2 w|| =
Acknowledgements • Unit Circle: http: //www. davidhardison. com/math/trig/unit_circle. gif • Text: Blitzer, Precalculus Essentials, Pearson Publishing, 2006.
- Asu mat 142
- Math 20-1 trigonometry review
- Ib
- Http //mbs.meb.gov.tr/ http //www.alantercihleri.com
- Siat ung sistem informasi akademik
- · meaning in math
- Hit the button
- Asu academic status report
- Asu.navs
- Asu mat 275
- Asu color palette
- Kawski asu
- Yasemin asu çırpıcı
- Asu cse 365
- Asu 2016 14
- Phy 121 asu
- Asu course catalogue
- Kevin salcido asu
- Asu mapps
- Blackboard astate
- Asu cse 340
- Brandguide asu
- Asu prep dress code
- Cse 545 asu github